Will Smith and his wife Jada Pinkett Smith are reportedly under investigation by child protective services.

A controversial photo showing their daughter Willow, 13, in bed with 20-year-old actor Moises Arias has sparked the alleged investigation.

Radar Online reports the Los Angeles Department of Children and Family Services has opened an investigation, and so far the family is being co-operative.

A version of the controversial Instagram photograph that Kylie Jenner posted on Tumblr. Photo: Kylie Jenner/Tumblr

''The investigation was formally opened last week and is being taken very seriously by the department,'' a source told Radar.

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''Will and Jada Pinkett Smith have been extremely co-operative with officials. Of course, they aren’t happy that their parenting skills are under scrutiny, but they understand.''

The black and white photo was posted on Instagram by Arias early in May and shows Willow lying in bed with a topless Arias sitting behind her, partially covered by sheets. It was also shared on their friend Kendall Jenner’s Tumblr account.

The post received a lot of criticism, and the Hannah Montana actor quickly removed it from his Instagram account.

Not long after it was posted, Jada Pinkett Smith lashed out at waiting paparazzi who asked her about the controversial photograph.

''Here’s the deal,'' Pinkett Smith said at Los Angeles Airport on May 7. ''There was nothing sexual about that picture or that situation. You guys are projecting your trash onto it. You’re acting like covert paedophiles and that’s not cool.''

TMZ reported sources close to the couple told them that Smith and his wife had ''no problem'' with the photograph at the time, as Willow is ''very mature'' for her age. Apparently the couple, who have been together for 15 years, saw the photograph as art and innocent fun.

Radar Online reports social workers will be interviewing the young Whip My Hair singer, and they also want to speak to Arias.

''This won’t just be one or two visits with the family,'' the source told Radar. ''And it will likely be an open investigation for at least a month, out of an abundance of caution.''

It is not the first time the couple have come under fire for their parenting style, but a formal investigation is certainly the most serious criticism.

Smith told Metro that he and his wife ''don’t do punishment''. ''The way that we deal with our kids is, they are responsible for their lives. Our concept is, as young as possible, give them as much control over their lives as possible,'' Smith told Metro. ''And the concept of punishment, our experience has been it has a little too much of a negative quality.''

Smith explained: ''So when they do things – and you know, Jaden, he’s done things – you can do anything you want as long as you can explain to me why that was the right thing to do for your life.''

Pinkett Smith has also acknowledged that the couple’s style of parenting might be a bit hard for some people to accept.

''How Will and I raise our family is a tremendous change from the norm,'' Pinkett Smith wrote on her Facebook page in 2012. ''It can be scary for some folks but it should never THREATEN how anyone else chooses to raise their own children or how they choose to live their lives.''